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Source: Manly, Bryan F.J. Multivariate Statistical Methods: A Primer, Third Edition, CRC Press,

07/2004.

Page 1 of 5

Tutorial Solutions– Week 4 (HT)

Question 1:

When you have several dependent variables and several samples/groups the four statistics

that may be used to identify differences between group means are Pillai’s trace, Wilk’s

Lambda, Roy’s largest root, and Lawes-Hotelling trace. Briefly, describe and compare.

Solution:

All have F equivalents. All compare some form of variation either within, between or total

SS.

Wilk’s: compares variation within groups to variation both within and between groups

(total) based on SS. A small Wilk’s indicates that the variation within is relatively small and

a significant difference between groups.

Roy’s looks at linear combination of variables that maximises ratio of between sample SS

and within sample SS. This ratio is lambda and the largest is the maximum latent root

(eigenvalue). A large lambda indicates a significant difference between groups.

Pillai’s trace also considers lambda and large lambda indicates a significant difference

between groups.

Lawes also considers lambda and large lambda indicates a significant difference between

groups.

Question 2:

How important are the assumption of MVN and equal covariances to Hotelling’s T2 and the

four statistics commonly used in MANOVA analysis?

Solution:

Chapter 4.2 Hotelling’s T2 is multivariate t-test. Some deviation from MVN is not too

important and moderate differences between population covariance matrices is fine.

All 4 MANOVA statistics assume MVN and equal covariance matrices, with Pillai’s trace the

most robust to deviations. All 4 tests are fairly robust to unequal sample sizes

(unbalanced).

Question 3:

When testing for normality is it possible to have non-significant univariate results and

significant multivariate results? Why?

Solution:

Chapter 4.4: Yes, because of accumulation of small-ish deviation from normality across

many variables can lead to significant deviation from MVN.

Question 4:

How does multiple testing affect Type I error rates? Explain how the Bonferroni correction

for multiple testing works.

Solution:

Chapter 4.4: multiple testing increases the chance of Type I error – reject Ho when samples

are not really from different populations. Specific multivariate tests like Hotelling’s T2 are an

advantage over series of univariate tests.

Source: Manly, Bryan F.J. Multivariate Statistical Methods: A Primer, Third Edition, CRC Press,

07/2004.

Page 2 of 5

Question 5:

Complete the exercise at the end of Chapter 4 of Manly. The data file ‘mandiblefull.dat’ is

available on the Study Desk. Some R code to get you started is provided in ‘mandible

MANOVA.R’

The variable names and codes are as follows:

X1 – length of mandible

X2 – breadth of mandible below 1st molar

X3 – breadth of articular condyle

X4 – height of mandible below 1st molar

X5 – length of 1st molar

X6 – breadth of 1st molar

X7 – length of 1st to 3rd molar inclusive (1st to 2nd for cuon)

X8 – length from 1st to 4th premolar inclusive

X9 – breadth of lower canine

Sex – Male (1), Female (2), Unknown (0)

Group – Thai (modern) dogs (1), golden jackals (2), cuons (3), Indian wolves (4),

Thai (prehistoric) dogs (5).

Solution:

> Y<-cbind(X1, X2, X3, X4, X5, X6, X7, X8,X9)

> (cory<-round(cor(Y), digits=3))

X1 X2 X3 X4 X5 X6 X7 X8 X9

X1 1.000 0.826 0.855 0.798 0.907 0.852 0.759 0.949 0.883

X2 0.826 1.000 0.786 0.890 0.821 0.846 0.560 0.746 0.887

X3 0.855 0.786 1.000 0.769 0.779 0.720 0.478 0.727 0.748

X4 0.798 0.890 0.769 1.000 0.740 0.809 0.471 0.715 0.823

X5 0.907 0.821 0.779 0.740 1.000 0.854 0.742 0.878 0.883

X6 0.852 0.846 0.720 0.809 0.854 1.000 0.646 0.798 0.894

X7 0.759 0.560 0.478 0.471 0.742 0.646 1.000 0.787 0.648

X8 0.949 0.746 0.727 0.715 0.878 0.798 0.787 1.000 0.838

X9 0.883 0.887 0.748 0.823 0.883 0.894 0.648 0.838 1.000

> mf.manova1<-manova(Y ~ as.factor(Group), data=mf)

> summary(mf.manova1) #default test is Pillai’s

Df Pillai approx F num Df den Df Pr(>F)

as.factor(Group) 4 2.5892 13.662 36 268 < 2.2e-16 ***

Residuals 72

—

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> summary(mf.manova1, test=”Wilks”)

Df Wilks approx F num Df den Df Pr(>F)

as.factor(Group) 4 0.0021936 27.666 36 241.57 < 2.2e-16 ***

Residuals 72

—

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> summary(mf.manova1, test=”Roy”)

Df Roy approx F num Df den Df Pr(>F)

as.factor(Group) 4 16.348 121.7 9 67 < 2.2e-16 ***

Residuals 72

—

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> summary(mf.manova1, test=”Hotelling-Lawley”)

Df Hotelling-Lawley approx F num Df den Df Pr(>F)

as.factor(Group) 4 25.129 43.627 36 250 < 2.2e-16 ***

Residuals 72

Source: Manly, Bryan F.J. Multivariate Statistical Methods: A Primer, Third Edition, CRC Press,

07/2004.

Page 3 of 5

—

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation between variables ranges from moderate (~0.4) to very high (>0.9). Some

correlation is needed for MANOVA however, it is possible that some of these highly

correlated variables should be removed from the analysis.

There is a significant difference (p<0.001) between at least two species in the mean ‘size’

of mandible (measured by 9 variables) based on all 4 tests.

> #to subset and run individual comparisons in MANOVA:

> mf.manova2<-manova(Y ~ as.factor(Group), data=mf,

+ subset=as.factor(Group)%in% c(“5”, “1”))

> summary(mf.manova2)

Df Pillai approx F num Df den Df Pr(>F)

as.factor(Group) 1 0.83288 8.8603 9 16 0.0001013 ***

Residuals 24

—

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> ##To run Hotellings on only 2 groups

> library(DescTools)

> G51<-subset(mf, Group==”5″|Group==”1″)

> (HotellingsT2Test(cbind(X1, X2, X3, X4, X5, X6, X7, X8,X9) ~ Group, data=G51))

Hotelling’s two sample T2-test

data: cbind(X1, X2, X3, X4, X5, X6, X7, X8, X9) by Group

T.2 = 8.8603, df1 = 9, df2 = 16, p-value = 0.0001013

alternative hypothesis: true location difference is not equal to c(0,0,0,0,0,0,0,0,0)

When comparing just 5-prehistoric dogs and 1-modern dogs using either MANOVA or

Hotelling’s T2 the two species are significantly different in ‘size’ (p<0.001). Notice the

difference in df now that we are only comparing 2 groups.

Remember: if you were to do multiple pairwise comparisons between species (Group) you

would need to control for Type I error rate and adjust the significance cut-off, e.g.

Bonferroni correction (divide p by the number of tests). See Manly Chapter 4.4.

> mf.manova3<-manova(Y ~ as.factor(Group), data=mf,

+ subset=as.factor(Group)%in% c(“5”, “2”))

> summary(mf.manova3)

Df Pillai approx F num Df den Df Pr(>F)

as.factor(Group) 1 0.9137 23.527 9 20 9.423e-09 ***

Residuals 28

—

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> mf.manova4<-manova(Y ~ as.factor(Group), data=mf,

+ subset=as.factor(Group)%in% c(“5”, “3”))

> summary(mf.manova4)

Df Pillai approx F num Df den Df Pr(>F)

as.factor(Group) 1 0.97745 81.894 9 17 3.222e-12 ***

Residuals 25

—

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> mf.manova5<-manova(Y ~ as.factor(Group), data=mf,

+ subset=as.factor(Group)%in% c(“5”, “4”))

> summary(mf.manova5)

Source: Manly, Bryan F.J. Multivariate Statistical Methods: A Primer, Third Edition, CRC Press,

07/2004.

Page 4 of 5

Df Pillai approx F num Df den Df Pr(>F)

as.factor(Group) 1 0.91706 17.198 9 14 4.213e-06 ***

Residuals 22

—

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

The mean ‘size’ described by nine variables differs significantly (p<0.001) between

Prehistoric dogs and all other species, where alpha=0.05/4 = 0.0125 applying a Bonferroni

correction for multiple testing.

> mf_sex<-subset(mf, Sex==”1″|Sex==”2″& Group<5)

> mf.manova6<-manova(cbind(X1, X2, X3, X4, X5, X6, X7, X8, X9) ~ as.factor(Group) *

as.factor(Sex), data=mf_sex)

> summary(mf.manova6)

Df Pillai approx F num Df den Df Pr(>F)

as.factor(Group) 3 2.32644 20.3400 27 159 < 2.2e-16 ***

as.factor(Sex) 1 0.38789 3.5909 9 51 0.001597 **

as.factor(Group):as.factor(Sex) 3 0.58485 1.4261 27 159 0.093163 .

Residuals 59

—

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

The interaction between sex and Group (species) is only significant at p<0.1.

A significant interaction effect means that the mean ‘size’ depends on both gender and

species so that within at least one of the 4 species there is a difference between gender and

between at least two species the effect of gender is different. This could be further explored

with individual tests by species. From the test above both Group and Sex are significant

factors in describing differences between mean ‘size’ (p<0.001). When an interaction is

significant it is generally unwise to then interpret significant individual effects separately (if

there is a significant interaction then this means that one factor affects the other so it could

be misleading to interpret them independently). However in this case, the significance of

the interaction is not very convincing (only just significant at 0.1) so I would consider the

effects of both Group and Sex independently.

> G1<-subset(mf, Group==”1″ , select=c(X1, X2, X3, X4, X5, X6, X7, X8,X9))

> G5<-subset(mf, Group==”5″ , select=c(X1, X2, X3, X4, X5, X6, X7, X8,X9))

> par(mfrow=c(1,2))

> boxplot(G1, xlab=”size variables for modern dogs (1)”, ylab=”size (mm)”)

> boxplot(G5, xlab=”size variables for prehistoric dogs (2)”, ylab=”size (mm)”)

Source: Manly, Bryan F.J. Multivariate Statistical Methods: A Primer, Third Edition, CRC Press,

07/2004.

Page 5 of 5

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